500 gm of ice at – `5^(@)C` is mixed with 100 gm of water at `20^(@)C`when equilibrium is reached, amount of ice in the mixture is:

Options

A. 93.75 gm
B. 421.75 gm
C. 406.25 gm
D. zero

Explanation

To solve the problem of finding the amount of ice remaining after mixing 500 g of ice at -5°C with 100 g of water at 20°C, we can follow these steps:

### Step 1: Calculate the heat required to raise the temperature of ice from -5°C to 0°C.

We use the formula:
\[ Q_1 = m \cdot s \cdot \Delta T \]
Where:
- \( m = 500 \, \text{g} \) (mass of ice)
- \( s = 0.5 \, \text{cal/g°C} \) (specific heat of ice)
- \( \Delta T = 5 \, \text{°C} \) (change in temperature)

Calculating \( Q_1 \):
\[ Q_1 = 500 \, \text{g} \cdot 0.5 \, \text{cal/g°C} \cdot 5 \, \text{°C} = 1250 \, \text{cal} \]

### Step 2: Calculate the heat released by water when it cools from 20°C to 0°C.

Using the same formula:
\[ Q_2 = m \cdot s \cdot \Delta T \]
Where:
- \( m = 100 \, \text{g} \) (mass of water)
- \( s = 1 \, \text{cal/g°C} \) (specific heat of water)
- \( \Delta T = 20 \, \text{°C} \) (change in temperature)

Calculating \( Q_2 \):
\[ Q_2 = 100 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 20 \, \text{°C} = 2000 \, \text{cal} \]

### Step 3: Determine the net heat available for melting the ice.

The heat available for melting the ice is the difference between the heat released by the water and the heat required to raise the temperature of the ice:
\[ Q_{\text{net}} = Q_2 - Q_1 \]
\[ Q_{\text{net}} = 2000 \, \text{cal} - 1250 \, \text{cal} = 750 \, \text{cal} \]

### Step 4: Calculate the mass of ice that can be melted with the available heat.

The latent heat of fusion of ice is approximately 80 cal/g. We can find the mass of ice melted using:
\[ m = \frac{Q_{\text{net}}}{L} \]
Where \( L = 80 \, \text{cal/g} \).

Calculating the mass of melted ice:
\[ m = \frac{750 \, \text{cal}}{80 \, \text{cal/g}} = 9.375 \, \text{g} \]

### Step 5: Calculate the remaining amount of ice.

The initial mass of ice was 500 g. After melting some of it, the remaining mass of ice is:
\[ \text{Remaining ice} = 500 \, \text{g} - 9.375 \, \text{g} = 490.625 \, \text{g} \]

### Final Answer:
The amount of ice remaining in the mixture when equilibrium is reached is approximately **490.63 g**.

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